98edo

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98edo

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Prime factorization	2 × 72
Step size	12.2449¢
Fifth	57\98 (697.959¢)
Semitones (A1:m2)	7:9 (85.71¢ : 110.2¢)
Consistency limit	3
Distinct consistency limit	3

98 equal divisions of the octave (abbreviated 98edo or 98ed2), also called 98-tone equal temperament (98tet) or 98 equal temperament (98et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 98 equal parts of about 12.2 ¢ each. Each step represents a frequency ratio of 2^1/98, or the 98th root of 2.

Theory

The patent val of 98edo has a flat 3, a sharp 5 and a slightly flat 7, and tempers out 81/80 in the 5-limit, making it a system of meantone family with a 4-cent-flat fifth. In the 7-limit it tempers out 1029/1024, 1728/1715, supporting mothra temperament, in the 11-limit 176/175 and 540/539, supporting mosura, and in the 13-limit 144/143 and 196/195. It provides the optimal patent val for 13-limit mosura temperament.

Odd harmonics

Approximation of odd harmonics in 98edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-4.00	+5.52	-1.48	+4.25	-0.30	+4.37	+1.53	+5.25	-3.64	-5.47	-3.78
	Relative (%)	-32.6	+45.1	-12.1	+34.7	-2.4	+35.7	+12.5	+42.9	-29.7	-44.7	-30.9
Steps (reduced)		155 (57)	228 (32)	275 (79)	311 (17)	339 (45)	363 (69)	383 (89)	401 (9)	416 (24)	430 (38)	443 (51)

Subsets and supersets

Since 98 factors into 2 × 7², 98edo has subset edos 2, 7, 14, and 49.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	12.2		^D, vE♭♭
2	24.5		^^D, E♭♭
3	36.7		^3D, ^E♭♭
4	49	35/34	v3D♯, ^^E♭♭
5	61.2	29/28, 30/29	vvD♯, ^3E♭♭
6	73.5	24/23	vD♯, v3E♭
7	85.7	41/39	D♯, vvE♭
8	98	37/35	^D♯, vE♭
9	110.2	16/15	^^D♯, E♭
10	122.4	44/41	^3D♯, ^E♭
11	134.7	40/37	v3D𝄪, ^^E♭
12	146.9	37/34	vvD𝄪, ^3E♭
13	159.2	23/21, 34/31	vD𝄪, v3E
14	171.4	21/19, 32/29, 43/39	D𝄪, vvE
15	183.7		^D𝄪, vE
16	195.9		E
17	208.2	35/31, 44/39	^E, vF♭
18	220.4		^^E, F♭
19	232.7	8/7	^3E, ^F♭
20	244.9	38/33	v3E♯, ^^F♭
21	257.1		vvE♯, ^3F♭
22	269.4	7/6	vE♯, v3F
23	281.6	20/17	E♯, vvF
24	293.9		^E♯, vF
25	306.1	31/26, 37/31	F
26	318.4		^F, vG♭♭
27	330.6	23/19	^^F, G♭♭
28	342.9	28/23, 39/32	^3F, ^G♭♭
29	355.1	43/35	v3F♯, ^^G♭♭
30	367.3		vvF♯, ^3G♭♭
31	379.6		vF♯, v3G♭
32	391.8		F♯, vvG♭
33	404.1	24/19, 43/34	^F♯, vG♭
34	416.3	14/11	^^F♯, G♭
35	428.6	41/32	^3F♯, ^G♭
36	440.8	40/31	v3F𝄪, ^^G♭
37	453.1	13/10	vvF𝄪, ^3G♭
38	465.3	17/13	vF𝄪, v3G
39	477.6	29/22	F𝄪, vvG
40	489.8		^F𝄪, vG
41	502		G
42	514.3	35/26, 39/29	^G, vA♭♭
43	526.5	19/14	^^G, A♭♭
44	538.8	15/11, 41/30	^3G, ^A♭♭
45	551	11/8	v3G♯, ^^A♭♭
46	563.3		vvG♯, ^3A♭♭
47	575.5	39/28	vG♯, v3A♭
48	587.8		G♯, vvA♭
49	600	41/29	^G♯, vA♭
50	612.2	37/26	^^G♯, A♭
51	624.5	33/23, 43/30	^3G♯, ^A♭
52	636.7		v3G𝄪, ^^A♭
53	649	16/11	vvG𝄪, ^3A♭
54	661.2	22/15, 41/28	vG𝄪, v3A
55	673.5	28/19	G𝄪, vvA
56	685.7		^G𝄪, vA
57	698		A
58	710.2		^A, vB♭♭
59	722.4	44/29	^^A, B♭♭
60	734.7	26/17	^3A, ^B♭♭
61	746.9	20/13	v3A♯, ^^B♭♭
62	759.2	31/20	vvA♯, ^3B♭♭
63	771.4		vA♯, v3B♭
64	783.7	11/7	A♯, vvB♭
65	795.9	19/12	^A♯, vB♭
66	808.2		^^A♯, B♭
67	820.4		^3A♯, ^B♭
68	832.7		v3A𝄪, ^^B♭
69	844.9		vvA𝄪, ^3B♭
70	857.1	23/14	vA𝄪, v3B
71	869.4	38/23, 43/26	A𝄪, vvB
72	881.6		^A𝄪, vB
73	893.9		B
74	906.1		^B, vC♭
75	918.4	17/10	^^B, C♭
76	930.6	12/7	^3B, ^C♭
77	942.9		v3B♯, ^^C♭
78	955.1	33/19	vvB♯, ^3C♭
79	967.3	7/4	vB♯, v3C
80	979.6		B♯, vvC
81	991.8	39/22	^B♯, vC
82	1004.1		C
83	1016.3		^C, vD♭♭
84	1028.6	29/16, 38/21	^^C, D♭♭
85	1040.8	31/17, 42/23	^3C, ^D♭♭
86	1053.1		v3C♯, ^^D♭♭
87	1065.3	37/20	vvC♯, ^3D♭♭
88	1077.6	41/22	vC♯, v3D♭
89	1089.8	15/8	C♯, vvD♭
90	1102		^C♯, vD♭
91	1114.3		^^C♯, D♭
92	1126.5	23/12	^3C♯, ^D♭
93	1138.8	29/15	v3C𝄪, ^^D♭
94	1151		vvC𝄪, ^3D♭
95	1163.3		vC𝄪, v3D
96	1175.5		C𝄪, vvD
97	1187.8		^C𝄪, vD
98	1200	2/1	D

Music

Bryan Deister

• microtonal improvisation in 98edo (2023)